Generalized asymptotic Sidon basis

TL;DR

This paper proves the existence of special sets of positive integers called $B_h[1]$ sets that serve as asymptotic bases of order $2h+1$, using probabilistic methods to extend the theory of Sidon sets.

Contribution

It introduces the existence of $B_h[1]$ sets that are asymptotic bases of order $2h+1$, expanding the understanding of additive bases and Sidon sets.

Findings

Existence of $B_h[1]$ sets as asymptotic bases of order $2h+1$

Use of probabilistic methods to establish set existence

Extension of Sidon basis theory to generalized asymptotic bases

Abstract

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called $B_{h}[g]$ set if all positive integers can be represented as the sum of $h$ terms from $A$ at most $g$ times. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h+1$ by using probabilistic methods.